Master Thesis - Department of Computer Science

u
2
x 2
u 1
x 1
(a) PCA basis (b) PCA reduction to 1D
Figure 2.2: The concept of PCA. (a) Solid lines: The original basis; Dashed lines:
The PCA basis; Geometric interpretation of principal eigenvectors illustrated in 2D
space. (b) The projection (1D reconstruction) of the data using the first principal
component.
M is total number of pixels in the images and N is the total number of samples. Each
of the face images xi belongs to one of the C classes {1, 2, ....., C}.
• PCA (Principal Component Analysis): The key idea behind PCA [114,
124] is to find the best set of projection directions in the sample space that
maximizes total scatter across all images. This is accomplished by computing
a set of eigenfaces from the eigenvectors of total scatter matrix St, defined as:
x 2
N�
St = (xi − m)(xi − m)
i=1
T , (2.1)
where m is the mean face of the sample set X. The geometric interpretation of
PCA is shown in Fig. 2.2. For dimensionality reduction, K (where K < M)
eigenvectors U = [u1, u2, ..., uK] corresponding to first K largest eigenvalues
of St are selected as eigenfaces. Reduced dimension training samples, Y =
[y1, y2, ....., yN] can be obtained by the transformation Y = U T X. Now, when
a probe image xt is presented for identification/verification, it is projected on
U to obtain a reduced vector yt = U T xt. A response vector of length C,
R(xt) = [r1, r2, . . . , rC] is calculated by measuring distances from the probe to
the nearest training samples from each class. The distance function between
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u 1
x 1